Papercraft Polyhedra

Drawing and constructing polyhedra is a pastime that goes back to the Renaissance and perhaps even earlier times. Leonardo da Vinci (1452–1519), for one, created illustrations of various polyhedra for a 1509 book on the divine proportion by Luca Pacioli (1445–1517).

These immensely varied, crystal-like shapes, with regular features and flat faces (plane polygons), come in all sorts of configurations. Many people know of the five regular polyhedra (Platonic solids): the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. But the realm of polyhedra encompasses all sorts of additional forms: spiky stellated polyhedra, intricate, interlocked shapes, buckyballs and their cousins, and many more.

Fr. Magnus J. Wenninger (1919-2017), a mathematician and philosopher at Saint John's Abbey in Collegeville, Minn., spent much of his life (since 1961) painstakingly and meticulously constructing polyhedra. His colorful, precise models, fashioned from paper and typically ranging from 30 to 40 centimeters in diameter, reflected the broad range of shapes that symmetrical polyhedra can take on. 

Fr. Magnus Wenninger in 2005 with one of his larger polyhedral models, called "Fifteen Cubes."
Courtesy of Fr. Magnus Wenninger.

Creating such models was no simple task. I had a chance to observe Father Magnus quietly at work several times at various conferences on mathematics and art. His patience, care, and skill were clearly evident. And the results were awesome.

A sampling of Fr. Magnus Wenninger's intricate, precise polyhedral models.

Courtesy of Fr. Magnus Wenninger.

Over the years, Father Magnus wrote numerous articles and several books about how to construct accurate models of various types of polyhedra. Later, he worked with other polyhedron experts to develop design software for creating polyhedral forms.

If you're interested in templates and patterns for such forms, software such as Stella, developed by Robert Webb, provides a good starting point.

Originally posted April 23, 2006

Updated February 20, 2017

References

Wenninger, M. 1999. Spherical Models. Dover.

______. 1983. Dual Models. Cambridge University Press.

______. 1971. Polyhedron Models. Cambridge University Press.

Pentium Bug Revisited

The jokes still circulate on the Internet:

Q: How many Pentium designers does it take to screw in a light bulb?
A: 1.99995827903, but that's close enough for nontechnical people.

At Intel, quality is job 0.999999998.

Q. What do you get when you cross a Pentium PC with a research grant?
A. A mad scientist.

The incident that triggered this barrage of ridicule occurred in the fall of 1994, a few months after Intel had introduced its Pentium microprocessor. The furor started with an e-mail message from Thomas R. Nicely, a mathematician at Lynchburg College in Virginia, who was doing computations related to the distribution of prime numbers. Nicely pointed out that the chip gave incorrect answers to certain floating-point (decimal) division calculations. Other users quickly confirmed the problem and identified additional examples in which an error occurs.

Here's one of the more dramatic instances:
x = 4195835, y = 3145727, and z = x - (x/y)*y.

Computed exactly, the answer, z, should be zero. The Pentium chip gave 256 as the answer.

Typically, the flawed chip generated a slightly inaccurate answer for only a few pairs of numbers. For example, dividing 5505001 by 294911 produced 18.66600093 instead of 18.66665197. Multiplying (or dividing) either number of the pair by any power of 2 also gave an incorrect answer. Intel claimed that only one in nine billion division operations would exhibit reduced precision.

So, the probability of randomly coming across the affected numbers was very small. But computations don't necessarily involve random selections of numbers, and such errors can matter in scientific and engineering applications and even in some spreadsheet calculations.

The fault itself had occurred because of the omission of five entries in a table of 1,066 values (part of the chip's circuitry) used by a speedy algorithm known as SRT division, after the initials of the three researchers who independently introduced the scheme more than 30 years ago. The five cells should have contained the constant +2, but because the cells were empty, the processor fetched a zero. This could throw off a calculation, which would result in a number that was always slightly less precise than the correct answer.

At risk were the binary divisors beginning with 1.0001, 1.0100, 1.0111, 1.1010, and 1.1101. These particular bit patterns had to be followed by a string of six 1s to boost the probability of error.

Intel corrected the problem and a variety of other glitches, and the Pentium chip (in all its variants) is now the workhorse of the desktop information age. However, that's not really the end of the story.

"It turns out the mathematics is more interesting than people think," says Alan Edelman, an applied mathematician at the Massachusetts Institute of Technology. "Most people link [the Pentium bug] to a simple arithmetic mistake, but it is far more subtle."

Edelman describes some of these nuances in a paper published in the March 1997 SIAM Review. "The bug in the Pentium was an easy mistake to make, and a difficult one to catch," he says.

The radix-4 SRT algorithm used on the Pentium chip represents a variation of the long-division method we learned in school. Suppose we want to divide 63 into 819762: 63 goes into 81 once, leaving a remainder of 18; then 9 is appended to the remainder to produce 189, and 63 divides into this number 3 times, leaving a remainder of zero. So far, the quotient is 13. The same procedure is repeated until the computation is complete.

The SRT algorithm does essentially the same thing, except that it works with the numbers -2, -1, 0, 1, and 2 instead of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 -- that is, in base 4 instead of base 10. Its advantage in a computer is that each step adds another 2-bit digit to the quotient, speeding up the rate at which division can be performed.

The Pentium chip's table, implemented as a triangular array, allowed it to look up values required for doing the computation. "When the algorithm was implemented in silicon, five entries on the boundary of this triangular portion were dropped," says Cleve Moler of MathWorks, Inc., who worked out a method for circumventing the problem. "These missing entries . . . effectively mean[s] that, for certain combinations of bits developed during the division process, the chip makes an error while updating the remainder."

Edelman's analysis reveals that a buggy value can be reached only from the entry below it in the chip's table. "This entry just below the bad entry plays the role of a foothold," he says. "The existence of this foothold is a subtle phenomenon that may not have been readily guessed from general properties of SRT division.

"I believe that its existence has surprised everybody," Edelman continues. "Any thought that each table entry is somehow equally likely to be reached is very wrong."

In effect, there are only two viable routes to arrive at a bug. Even then, it is quite possible to reach the foothold entry yet not hit the buggy entry on the next step. So, errors in the final output would occur only under very specific circumstances.

"Naively, one might have thought that the bad table entries would be hit uniformly at random," Edelman observes. "If this had been the case, the bug surely would have been noticed earlier and would have had the potential to cause more significant damage in every instance it went unnoticed."

Now, the Internet is buzzing with reports of a bug involving floating-point arithmetic in the new Pentium II microprocessor. Floating-point numbers are stored inside the microprocessor as strings of 80 bits, whereas integers are stored in 16 or 32 bits. Sometimes a floating-point number converted to an integer doesn't fit the smaller format, and the microprocessor is supposed to provide an "overflow" warning. Apparently, the Pentium II chip fails to do so in many cases.

Interestingly, the launch failure of the Ariane 5 rocket, which exploded 37 seconds after liftoff on June 4, 1996, occurred because of a software error that resulted from converting a 64-bit floating point number to a 16-bit integer. The value of the floating point number happened to be larger than could be represented by a 16-bit integer. The overflow wasn't handled properly, and in response, the computer cleared its memory. The memory dump was interpreted by the rocket as instructions to its rocket nozzles, and an explosion resulted.
It doesn't take much!

Originally posted May 12, 1997

References:

Cipra, Barry. 1995. How number theory got the best of the Pentium chip. Science 267(Jan. 13):175.

Coe, Tim, et al. 1995. Computational aspects of the Pentium affair. IEEE Computational Science & Engineering (Spring):18-30.

Edelman, Alan. 1997. The mathematics of the Pentium division bug. SIAM Review 39(March):54-67. Available at http://www.siam.org/journals/sirev/39-1/29395.html.

Fisher, Lawrence M. 1997. Flaw reported in new Intel chip. New York Times (May 5). Available at http://www.nytimes.com/library/cyber/week/050697intel-chip-flaw.html

Geppert, Linda. 1995. Biology 101 on the Internet: Dissecting the Pentium bug. IEEE Spectrum (February):16-17.

Moler, Cleve. 1995. A tale of two numbers. SIAM News 28(January):1.

Nuseibeh, Bashar. 1997. Ariane 5: Who dunnit? IEEE Software 14(May/June):15-16.

Peterson, Ivars. 1996. Fatal Defect: Chasing Killer Computer Bugs. New York: Vintage.

Losing to Win

It's a gift to born losers. Researchers have demonstrated that two games of chance, each guaranteed to give a player a predominance of losses in the long term, can add up to a winning outcome if the player alternates randomly between the two games.

This striking result in game theory is now called Parrondo's paradox, after its discoverer, Juan M.R. Parrondo, a physicist at the Universidad Complutense de Madrid in Spain.

A combination of two losing gambling games illustrates this counterintuitive phenomenon. The two games involve tossing biased coins. In the simpler game, the player gambles with a coin that's been loaded to make the probability of winning less than 50 percent. Winning means that the player receives $1 and losing means that the player loses $1 on each turn.

GAME 1

Probability of winning: ½ - α

Probability of losing: ½ + α

The second, more complicated game requires two biased coins. One of the coins wins more often than it loses, and the other loses more often than it wins. The game is set up so that even though the winning coin is tossed more often, this is outweighed by the much lower probability of winning with the other coin.

Here's the rule for the two coins in the second game. If the player's total amount of cash on hand is a multiple of 3, the chance of winning is just 1/10 – α. If not, the chance of winning is higher: ¾ - α.

GAME 2

Is the total amount of cash on hand a multiple of 3?

NO

Coin 2

Probability of winning: ¾ - α

Probability of losing: ¼ - α

YES

Coin 3

Probability of winning: 1/10 – α

Probability of losing: 9/10 + α

When α is greater than zero, each game played repeatedly on its own gradually depletes a player's capital.

However, if a player starts switching between the two games, playing two turns of game 1, then two turns of game 2, and so on, he or she starts winning. Randomly switching between the games also results in a steady increase in capital. Indeed, playing games 1 and 2 in any sequence leads to a win.

Gregory P. Harmer and Derek Abbott of the University of Adelaide in Australia ran computer simulations of the games, demonstrating this counterintuitive result for 50,000 trials at α = 0.005.

Alternating between the games produces a ratchet-like effect. Imagine an uphill slope with its steepness related to a coin's bias. Winning means moving uphill. In the single-coin game, the slope is smooth, and in the two-coin game, the slope has a sawtooth profile. Going from one game to the other is like switching between smooth and sawtooth profiles. In effect, any winnings that happen to come along are trapped by the switch to the other game before subsequent repetitions of the original game can contribute to the otherwise inevitable decline.

The same type of ratchet effect can occur in a bag or can of mixed nuts, Abbott says. Brazil nuts tend to rise to the top because smaller nuts block downward movement of the larger nuts.

"There are actually many ways to construct such gambling scenarios," Harmer and Abbott commented in the Dec. 23/30, 1999 Nature. The researchers suggested that similar strategies may operate in the economic, social, or ecological realms to extract benefits from what look like detrimental situations.

Unfortunately, Parrondo's paradox doesn't work for the types of games played in casinos.

Original version posted March 6, 2000

References:

Ball, P. 1999. Good news for losers. Nature Science Update (Dec. 23).

Blakeslee, S. 2000. Paradox in game theory: Losing strategy that wins. New York Times (Jan. 25).

Bogomolny, A. 2001. Parrondo paradox. Cut the Knot! (June).

Harmer, G.P., and D. Abbott. 1999. Losing strategies can win by Parrondo's paradox. Nature 402(Dec. 23/30): 864.

McClintock, P.V.E. 1999. Random fluctuations: Unsolved problems of noise. Nature 401(Sept. 2):24.

Peterson, I. 2000. Losing to win. Science News 157(Jan. 15):47.

Arithmagic

Watch him pull a rabbit out of his hat? Not exactly.

Arthur T. Benjamin eschews the usual trappings of the magician's trade. Calling himself a mathemagician, he astonishes audiences with amazing feats of mental arithmetic. Behind the scenes, he reveals how you, too, can look like a genius without really trying.

Photo by Richard Haverty

A math professor at Harvey Mudd College, Benjamin has brought his particular brand of prestidigitation to a wide variety of appreciative audiences. He has appeared on television programs and performed in night clubs, classrooms, and banquet halls. He has even traded quips with comedian Stephen Colbert on The Colbert Report.

Benjamin's act is a striking demonstration of how miraculous fairly ordinary, simple mathematical machinery can appear when you don't really know what's going on under the hood.

One of my first encounters—many, many years ago—with arithmetic shortcuts was in a book titled Cheaper by the Dozen, the story of efficiency expert Frank B. Gilbreth and his family of 12 children. In a chapter about dinner conversation, Frank B. Gilbreth Jr. and Ernestine Gilbreth Carey write, "Also of exceptional general interest was a series of tricks whereby Dad could multiply large numbers in his head, without using pencil and paper."

For example, to multiply 46 times 46, you figure out how much greater 46 is than 25. The answer is 21. Then you calculate how much less 46 is than 50. The answer is 4. You square 4 and get 16. Putting 16 and 21 together gives the answer 2116.

That's the sort of venerable procedure that Benjamin uses to do fast multiplies and to square four-digit numbers faster than someone using a calculator. Turning a repertoire of such calculating tricks into a real show, however, requires developing a memory for numbers and learning how to calculate from left to right, performed at the speed of rapid-fire chatter.

Suppose you want to multiply 378 by 7. Starting from the left, you would get 2100 (300 x 7) plus something more. In the next step, 70 x 7 equals 490, which is added to 2100 to give 2590, plus something more. Finally, 7 x 8 equals 56, which is added to 2590 to give 2646.

One advantage of using this method is that you can start saying the answer while you're still calculating it, Benjamin remarks.

Here's a neat way to square two-digit numbers. Suppose the number to be squared is 37. That number is 3 less than 40, and 34 is 3 less than 37. Multiply 40 by 34 to get 1360, then add the square of the difference, 3² or 9, to get 1369.

The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3480, then add 1² , to obtain 3481.

The proof is in the algebra: a² = (a + d)(a - d) + d² . The same idea can be extended to squaring 3-digit and 4-digit numbers.

Interestingly, when Benjamin performs calculations involving long sequences of digits, he relies on a phonetic code to remember the numbers. "There's no mental blackboard," he says. "It's much more an auditory process."

Benjamin turns sequences of digits into words that add up to some sort of crazy scenario. For example, he can mentally convert the sequence 9 6 4 8 3 7 5 4 8 3 1 2 7 5 9 6 into the words "pitcher fume color fume tinkle beach" as a handy mnemonic for (9 6 4) (8 3) (7 5 4) (8 3) (1 2 7 5) (9 6).

The underlying scheme assigns different consonants to different numbers, and the memorizer supplies the vowels: 1 (t, d), 2 (n), 3 (m), 4 (r), 5 (l), 6 (j, ch, sh), 7 (c, k, g), 8 (f, v, ph), 9 (p, b), 10 (z, s). This is the modern version of a number alphabet originally proposed by Pierre Hérigone and published in Paris in 1634. Three consonants—w, h, and y, spelling "why"—do not appear in the list.

For more on phonetic codes, see Benjamin’s article "A Better Way to Memorize Pi: The Phonetic Code," published in the February 2000 Math Horizons.

Benjamin can also handle magic squares, natural logarithms, cube roots, and much more. "As a kid, I liked to show off," he says. "Now I get paid to do this!" Besides, the techniques are so easy that even an elementary-school student can master them.

Benjamin describes and explains many of his mental calculation techniques in the book Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks (coauthored by Michael Shermer).

He also has a set of DVDs in The Great Courses series devoted to "Secrets of Mental Math." Benjamin’s "Mathemagic" performance, available as a TED video, has been viewed more than 5 million times.

Benjamin wrote about the mathematics underlying some of his mental feats in an article published in the January 2012 College Mathematics Journal—a special issue dedicated to Martin Gardner. Titled "Squaring, Cubing, and Cube Rooting," the article starts off:

"I still recall the thrill and simultaneous disappointment I felt when I first read Mathematical Carnival by Martin Gardner. I was thrilled because, as my high school teacher had told me, mathematics was presented there in a playful way that I had never seen before. I was disappointed because Gardner quoted a formula that I thought I had 'invented' a few years earlier. I have always had a passion for mental calculation, and the formula . . . appears in Gardner's chapter on 'lightning calculators.' It was used by the mathematician A. C. Aitken to square large numbers mentally."

Benjamin's article is reprinted in Martin Gardner in the Twenty-First Century, edited by Michael Henle and Brian Hopkins (MAA, 2012).

Original version posted October 5, 1998

Reference:

Gardner, M. 2008. Memorizing numbers. In Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games. Cambridge University Press and MAA.

A Lawyer's Math Library

"Strangely enough, anyone wishing to write about Galois in Paris would do well to journey to Louisville, Kentucky."—Leopold Infeld, Whom the Gods Love

French mathematician Evariste Galois (1811-1832), whose death in a duel at the age of 20 cut short a remarkably productive career, is just one of many mathematicians represented in a little-known collection of rare mathematical and astronomical books at the University of Louisville library.

A visitor can leaf through the wrinkled, yellowed pages, stiff with age, of the first printed edition of Euclid's Elements (Elementarum Euclidis) (1482), through Narratio Prima, in which Copernicus's pupil Georg Rheticus (1514-1576) announced the Copernican sun-centered concept of the solar system, and through a copy of Isaac Newton's Principia, with Newton's own handwritten corrections on the errata leaf.

The man who assembled this notable collection was an attorney and a mathematics enthusiast. Born in 1873 into a prominent Kentucky family, William Marshall Bullitt throughout his long life believed firmly in the value of mathematics.

Lurline Jochum, Bullitt's secretary from 1927 until his death in 1957, once recalled, "When a young man from law school would come into the office and want a job, the first thing [Bullitt] would say is: How much mathematics have you had? He felt that if you had a good mathematical background, then you had a good reasoning power."

While an undergraduate at Princeton University, Bullitt himself took mathematics courses in preparation for his subsequent legal career. Later, he studied at the University of Louisville law school and established a lucrative practice in Louisville, specializing in actuarial and constitutional law. His clients included several of the country's largest insurance companies. He even came up with a mathematical formula that helped him win several insurance cases, beginning with an important case for the New York Life Insurance Company.

Bullitt also served as Solicitor General of the United States for a brief period under President Taft.

At the same time, Bullitt kept up with developments in mathematics and astronomy by attending meetings of the American Mathematical Society and other groups and by corresponding with mathematicians and scientists, including Albert Einstein (1879-1955). His friends included astronomer Harlow Shapley (1885-1972) and mathematicians George D. Birkhoff (1884-1944), E.T. Bell (1883-1960), and Richard Courant (1888-1972).

Bullitt's goal of collecting "the most important original works of the most prominent mathematicians of all time" was established during a parlor game instigated by his friend, the prominent mathematician G.H. Hardy (1877-1947).

Like everything else he did, Bullitt went about his new project systematically. He asked Bell, Shapley, and others for lists of what they considered to be the most important books that he could collect. He wrote to mathematicians at various colleges all over the United States to get their comments on the lists. When he was ready, he notified rare-book dealers of his needs and even traveled personally to Germany and France to locate many of the works on his final list.

Starting his project in 1936, Bullitt didn't miss much in gathering first-edition works by the greatest mathematicians of all time. His final purchase for the collection, Niels H. Abel's 1824 Mémoire sur les Équations Algébriques, occurred in 1951. He paid $500—a sum he termed "outrageous."

Bullitt kept most of his collection in his law office, locking away some of the more valuable books in the office vault. In addition, he maintained a good selection of mathematics books in a magnificent library at Oxmoor, his family home located just outside of Louisville.

Visitors to Oxmoor can remember browsing through the library's mathematics books and Bullitt's habit of sometimes testing his visitors by posing mathematical puzzles.

One special feature of the collection attracted a few scholars even when Bullitt was still alive. Bullitt managed to assemble the most complete collection of the works of Galois to be found outside of France. This included copies of hard-to find, contemporary newspaper clippings, many unpublished items, and other documents.

When University of Toronto physicist Leopold Infeld (1898-1968) decided to write a biography of Galois, he visited Oxmoor and spent several days examining the collection. Infeld, a socialist, later described the visit—his first encounter with an American millionaire and the accompanying lifestyle—in his autobiography, Why I Left Canada.

"I still remember that in the bathroom the toilet paper was rose-colored and perfumed," Infeld wrote. "The window frames creaked so much in the wind that I was unable to sleep in the midst of all the abundance and luxury."

When Bullitt died, his widow donated the more valuable books to the University of Louisville, although schools such as Harvard would have liked to obtain the collection. Later, the remainder of the collection also went to the university library, and the current checklist contains about 370 items.

The collection is very rich in the authors that it covers, and it includes some extremely rare items. At the same time, most of the material is available elsewhere to mathematicians and interested historians in other forms or later editions.

Such is a resource is useful, however, when historians want to check original editions of mathematical works. In later editions, particularly during the 19th century, changes made by editors often obscured an author's original intent.

The William Marshall Bullitt Collection of Rare Mathematics and Astronomy at the University of Louisville Ekstrom Library gives visitors a chance to trace the mathematical formulas and geometrical diagrams of ancient authors, to puzzle out cryptic Greek and Latin phrases, and to contemplate some of the greatest achievements in mathematics. It affords an opportunity to touch a heritage.

Original version posted May 20, 2002

References:

Davitt, R.M. 1989. William Marshall Bullitt and his amazing mathematical collection. Mathematical Intelligencer 11(No. 4):26-33.

Infeld, L. 1948. Whom the Gods Love: The Story of Evariste Galois. National Council of Teachers of Mathematics.

______. 1978. Why I Left Canada: Reflections on Science and Politics. McGill-Queen's University Press.

Peterson, I. 1984. Off the beat: A lawyer's library. Science News 125(March 17):168-169.

A Song about Pi

Irving "Kap" Kaplansky was a prominent mathematician—a leading algebraist—who died in 2006.

I met Kaplansky in 1999, when he was 82 and still actively engaged in mathematical research. At that time, he was Director Emeritus of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., where I was spending the summer as Journalist in Residence.

Kaplansky spent much of his time then in the MSRI library, poking into various nooks and crannies of mathematical history. Tidying up loose ends and filling in gaps in the mathematical literature, he patiently worked through mathematical arguments, proved theorems, and prepared papers for publication. His remarkably wide-ranging efforts belied the oft-repeated notion that mathematicians are most productive when they are young.

A distinguished mathematician who made major contributions to algebra and other fields, Kaplansky was born in Toronto, Ontario, several years after his parents had emigrated from Poland. In the beginning, his parents thought that he was going to become a concert pianist. By the time he was 5 years old, he was taking piano lessons. That lasted for about 11 years, until he finally realized that he was never going to be a pianist of distinction.

Nonetheless, Kaplansky loved playing the piano, and music remained a lifelong hobby. "I sometimes say that God intended me to be the perfect accompanist—the perfect rehearsal pianist might be a better way of saying it," he said. "I play loud, I play in time, but I don't play very well."

While in high school, Kaplansky started to play in dance bands. During his graduate studies at Harvard, he was a member of a small combo that performed in local night clubs. For a while, he hosted a regular radio program, where he played imitations of popular artists of the day and commented on their music.

A little later, when Kaplansky became a math instructor at Harvard, one of his students was Tom Lehrer, later to become famous for his witty ditties about science and math (see "Tom Lehrer’s Derivative Ditties" for several examples).

In 1945, Kaplansky moved to the University of Chicago, where he remained until 1984, when he retired, then became MSRI director.

Songs had always interested him, particularly those of the period from 1920 to 1950. These songs tended to have a particular structure: the form AABA, where the A theme is repeated, followed by a contrasting B theme, then a return to the original A theme.

Early on, Kaplansky noticed that certain songs have a more subtle, complex structure. This alternative form can be described as AA'BAA'(B/2)A", where A is a four-bar phrase, A' and A" are variants, and B is a contrasting eight-bar phrase. "I don't think anyone had noticed that before," he said.

Kaplansky's discovery is noted in a book about the American musical by the late Chicago film scholar Gerald Mast.

Kaplansky argued that the second structure is really a superior form for songs. To demonstrate his point, he once used it to turn an unpromising source of thematic material—the first 14 decimal digits of pi—into a passable tune. In essence, each note of the song's chorus corresponds to a particular decimal digit.

When Chicago colleague Enid Rieser heard the melody at Kaplansky's debut lecture on the subject in 1971, she was inspired to write lyrics for the chorus.

A SONG ABOUT PI

Through all the bygone ages,
Philosophers and sages
Have meditated on the circle's mysteries.
From Euclid to Pythagoras,
From Gauss to Anaxag'ras,
Their thoughts have filled the libr'ies bulging histories.
And yet there was elation
Throughout the whole Greek nation
When Archimedes did his mighty computation!
He said:

CHORUS

3 1 41 Oh (5) my (9), here's (2) a (6) song (5) to (3) sing (5) about (8,9) pi (7).
Not a sigma or mu but a well-known Greek letter too.
You can have your alphas and your great phi-bates, and omegas for a friend,
But that's just what a circle doesn't have—the beginning or an end.
3 1 4 1 5 9 is a ratio we don't define;
Two pi times radii gives circumf'rence you can rely;
If you square the radius times the pi, you will get the circle's space.
Here's my song about pi, fit for a mathematician's embrace.

The chorus is in the key of C major, and the musical note C corresponds to 1, D to 2, and so on, in the decimal digits of pi.

You can hear a performance of the song by singer-songwriter Lucy Kaplansky (Irving Kaplansky's daughter) on YouTube. A club headliner, recording artist, and former psychologist, Lucy Kaplansky has her own distinctive style but doesn't mind occasionally showcasing her father's old-fashioned tunemanship.

In 1993, Irving Kaplansky wrote new lyrics for the venerable song "That's Entertainment!" to celebrate his enthusiasm for mathematics. He dedicated the verses to Tom Lehrer.

THAT'S MATHEMATICS

The fun when two parallels meet
Or a group with an action discrete
Or the thrill when some decimals repeat,
That's mathematics.
A nova, incredibly bright,
Or the speed of a photon of light,
Andrew Wiles, proving Fermat was right,
That's mathematics.
The odds of a bet when you're rolling two dice,
The marvelous fact that four colors suffice,
Slick software setting a price,
And the square on the hypotenuse
Will bring us a lot o' news.
In genes a double helix we see
And we cheer when an algebra's free
And in fact life's a big PDE.
We'll be on the go
When we learn to grow with mathematics.

With Lagrange everyone of us swears
That all things are the sums of four squares,
Like as not, three will do but who cares.
That's mathematics.
Sporadic groups are the ultimate bricks,
Finding them took some devilish tricks,
Now we know--there are just 26.
That's mathematics.
The function of Riemann is looking just fine,
It may have its zeros on one special line.
This thought is yours and it's mine.
We may soon learn about it
But somehow I doubt it.
Don't waste time asking whether or why
A good theorem is worth a real try,
Go ahead--prove transcendence of pi;
Of science the queen
We're all of us keen on mathematics.

Original version posted July 12, 1999

References:

Albers, D.J., G.L. Alexanderson, and C. Reid. 1990. Irving Kaplansky. In More Mathematical People: ContemporaryConversations. Academic Press.

Mast, G. 1987. Can't Help Singin': The American Musical on Stage and Screen. Overlook Press.

Kaplansky, I. 1992. The deep young man. Mathematical Intelligencer 14(No. 4):62.

An online video of Irving's Kaplansky's lecture on "Fun with mathematics: Some thoughts from seven decades" is available at http://www.msri.org/realvideo/ln/msri/1999/misc/kaplansky/1/index.html.

In the interest of full disclosure, I should note that I went to the same high school (Harbord Collegiate in Toronto) as Kaplansky and also attended the University of Toronto, though my schooling occurred a generation later.

Cubes of Perfection

Playing with integers can lead to all sorts of little surprises.

A whole number that is equal to the sum of all its possible divisors—including 1 but not the number itself—is known as a perfect number. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 equals 6.

Six is the smallest perfect number. The next is 28. Its proper divisors are 1, 2, 4, 7, and 14, and the sum of those divisors is 28.

Incidentally, if the sum works out to be less than the number itself, the number is said to be defective (or deficient). If the sum is greater, the number is said to be abundant. There are far more defective and abundant numbers than perfect numbers. Do abundant numbers actually outnumber defective numbers? I'm not sure.

Steven Kahan, a mathematics instructor at Queens College in Flushing, N.Y., has a long-standing interest in number theory and recreational mathematics. "I often play with number patterns," he says.

In the course of preparing a unit on number theory for one of his classes, he noticed a striking pattern involving the perfect number 28: 28 = 1³ + 3³. He tried the next perfect number: 496 = 1³ + 3³ + 5³ + 7³.

More than 2,000 years ago, the Greek geometer Euclid of Alexandria (325-265 B.C.) proved that if 2^p - 1 is a prime number, then 2^{p - 1}(2^p - 1) is an even perfect number. Primes of the form 2^p - 1 are now known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime.

Leonhard Euler (1707- 1783) proved the converse of Euclid's theorem: All even perfect numbers must have the form specified by Euclid's formula. Hence, every Mersenne prime automatically leads to a new perfect number. There are, at present, 48 known Mersenne primes.

It turns out that a given perfect number 2^{p - 1}(2^p - 1) greater than 6 is expressible as the sum of the cubes of the first n consecutive odd integers, where n = 2^{(p - 1)/2}. For example, if p = 7, then n = 8, and the perfect number 8,128 equals the sum of the cubes of the first eight odd integers: 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³. Kahan provides a short proof of this venerable theorem in the April 1998 Mathematics Magazine.

This means that the largest known perfect number, which has 34,850,340 digits, is the sum of the cubes of the first 2^28942580 consecutive odd integers.

Perfect numbers also show other curious patterns. Add the digits of any perfect number greater than 6, then add the digits of the sum together, and so on, until only one digit remains. That final digit is always 1.

28: 2 + 8 = 10; 1 + 0 = 1

496: 4 + 9 + 6 = 19; 1 + 9 = 10; 1 + 0 = 1

8,128: 8 + 1 + 2 + 8 = 19; 1 + 9 = 10; 1 + 0 = 1

Here's another remarkable relationship. The sum of the inverses of the divisors of a perfect number (leaving out 1 but including the number itself) is also 1.

6: 1/2 + 1/3 + 1/6 = 1

28: 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1

496: 1/2 + 1/4 + 1/8 + 1/16 + 1/31 + 1/62 + 1/124 + 1/248 + 1/496 = 1

Interesting digit patterns and numerical relationships also arise when Mersenne primes and perfect numbers are written out in binary form. Mersenne primes, for example, consist of unbroken strings of consecutive 1s—57,885,161 of them in the case of the current record holder.

Here are the first four perfect numbers: 110, 11100, 111110000, 1111111000000. See a pattern?

Happy hunting in perfect territory!

Originally posted May 18, 1998

References:

Gullberg, J. 1997. Mathematics: From the Birth of Numbers. W.W. Norton.

Kahan, S. 1998. Perfectly odd cubes. Mathematics Magazine 71(April):131.

Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. Holt.

An introduction to Mersenne primes and perfect numbers can be found at http://www.utm.edu/research/primes/mersenne.shtml.

Beyond Ultra Large and Extra Small

Metric (SI) prefixes come in handy for expressing both large and small quantities. So we have gigawatt power systems and terabyte hard drives at the high end, and we have nanoscale electronics and femtosecond lasers at the low end.

Advances in scientific research and technology keep pushing us to new frontiers in the domain of metric prefixes.

In a 1998 article in Scientific American, Philip Morrison and Phylis Morrison described the escalation in the amount of stored knowledge since the days of the famed library in ancient Alexandria. The library probably shelved the equivalent of 50,000 books today. Counting the literature of China, India, Iraq, and Iran, the total number of distinct books available 2,000 years ago might have been about 100,000.

In 1945, a librarian at Harvard estimated that "worthwhile" printed material amounted to about 10 million books. With one byte as a unit of information representing roughly the equivalent of a single character of English text, it's possible to store that text as 10 trillion bytes, or 10 terabytes, of data. Images, drawings, photographs, audio and video recordings, and huge databases, however, require much larger amounts of storage space.

The U.S. Library of Congress now holds more than 34.5 million books and other print materials. Because it also has 3.3 million sound recordings, 13.4 million photographs, 5.4 million maps, and other byte-hungry material, however, the total store comes to a couple of petabytes (a million billion bytes).

One can plausibly estimate that fresh text, including newspapers, amounts to less than 100 terabytes annually. Recorded music, films, photographs (including family snapshots), and other materials add considerably to that accumulation.

"The biggest byte makers are the television stations of the world," the Morrisons wrote. "Although it is hard to correct for innumerable repeats, our best source puts their originality at one tenth of all they send out and so allots them under 100 petabytes annually."

To assess the totality of information production, one can also try to include ephemeral signals conveyed from one person to another. The sounds of telephone calls add up to some 1,000 petabytes worldwide, or a few exabytes. Face-to-face speech provides several more exabytes of data.

At this stage, the summing gets trickier. The Morrisons simply concluded, "No estimate of the eventual human store seems quite credible as yet."

In a 2011 online article in Science Express, Martin Hilbert and Priscila Lopez provided new estimates of the world’s technological capacity to store, communicate, and compute information. They estimated that in 2007 humankind was able to store 2.9 x 10²⁰ optimally compressed bytes, communicate nearly 2 x 10²¹ bytes, and carry out 6.4 x 10¹⁸ instructions per second on general-purpose computers.

These estimates put us in a realm of metric prefixes that relatively few people yet know. In 1991 as part of the International System of Units (SI), the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures) adopted new prefixes representing 10²¹, 10²⁴, 10^-21, and 10^-24. This means that 1,000 exabytes equals 1 zettabyte (ZB), and 1,000 zettabytes equals 1 yottabyte (YB).

I first encountered the new prefixes in 1993 when researchers measured voltages in a superconducting circuit so small that they had to use the term milliattovolt, where "atto" stands for 10^-18. The proper term is zeptovolt.

Here's the current table of official metric (SI) prefixes:

Power of 10	Prefix	Symbol	Power of 10	Prefix	Symbol
24	yotta	Y	-1	deci	d
21	zetta	Z	-2	centi	c
18	exa	E	-3	milli	m
15	peta	P	-6	micro	m
12	tera	T	-9	nano	n
9	giga	G	-12	pico	p
6	mega	M	-15	femto	f
3	kilo	k	-18	atto	a
2	hecto	h	-21	zepto	z
1	deka	da	-24	yocto	y

Many of the prefixes come from Greek and Latin words, often via French. "Zepto" is derived from the Latin septem, meaning 7, because this is the seventh prefix in the system of metric prefixes. The s was replaced by z to avoid confusion with the abbreviation for the second. The prefix "zetta" was coined to parallel "zepto." Similarly, "yocto" is derived from the Latin octo, meaning 8, and "yotta" parallels that term.

You can find a guide to units of measurements and their history at A Dictionary of Units of Measurement.

You might be interested to know that an attoparsec is a distance of about one inch (3.1 centimeters). The distance for Earth of the most remote object yet observed in the universe is about 125 yottameters (13.2 billion light-years). The diameter of the largest known galaxy is about 53 zettameters. At the other end, an atomic mass unit equals 1.66 yoctograms.

The Convert Auto page (a student project) offers a handy tool for converting from one unit of measurement to another in a wide range of fields.

Now you can really start talking ultra large and extra small.

Originally posted July 27, 1998

References:

Hilbert, M., and P. Lopez. The world’s technological capacity to store, communicate, and compute information. Science Express (Feb. 10, 2011).

Morrison, P., and P. Morrison. 1998. Wonders: The sum of human knowledge? Scientific American 279(July):115.

Peterson, I. 1993. Measuring superconductor magnetic noise. Science News 143(Jan. 16):37.